Quadratic integral
In mathematics, a quadratic integral is an integral of the form
It can be evaluated by completing the square in the denominator.
Positive-discriminant case
Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by
and
The quadratic integral can now be written as
The partial fraction decomposition
allows us to evaluate the integral:
The final result for the original integral, under the assumption that q > 0, is
Negative-discriminant case
In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in
is positive. Then the integral becomes
References
- Weisstein, Eric W. "Quadratic Integral." From MathWorld--A Wolfram Web Resource, wherein the following is referenced:
- Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.
- v
- t
- e
Calculus
- Binomial theorem
- Concave function
- Continuous function
- Factorial
- Finite difference
- Free variables and bound variables
- Graph of a function
- Linear function
- Radian
- Rolle's theorem
- Secant
- Slope
- Tangent
- Indeterminate form
- Limit of a function
- Limit of a sequence
- Order of approximation
- (ε, δ)-definition of limit
- Derivative
- Second derivative
- Partial derivative
- Differential
- Differential operator
- Mean value theorem
- Notation
- Rules of differentiation
- Other techniques
- Stationary points
- Further applications
- Differential equation
- Derivatives
- Basic theorems
- Arithmetico-geometric sequence
- Types of series
- Tests of convergence
and numbers
- Bernoulli numbers
- e (mathematical constant)
- Exponential function
- Natural logarithm
- Stirling's approximation
- Differentiation rules
- List of integrals of exponential functions
- List of integrals of hyperbolic functions
- List of integrals of inverse hyperbolic functions
- List of integrals of inverse trigonometric functions
- List of integrals of irrational functions
- List of integrals of logarithmic functions
- List of integrals of rational functions
- List of integrals of trigonometric functions
- List of limits
- Lists of integrals
- Complex calculus
- Differential geometry
- Euler–Maclaurin formula
- Gabriel's horn
- Integration Bee
- Proof that 22/7 exceeds π
- Regiomontanus' angle maximization problem
- Steinmetz solid